[Paper Review] SU(2) Skyrme Model for Hadron

Miftachul Hadi, Hans J. Wospakrik|arXiv (Cornell University)|Jul 6, 2010

Quantum Chromodynamics and Particle Interactions6 references3 citations

TL;DR

This paper formulates the SU(2) Skyrme model as a soliton-based description of nucleons and delta baryons, treating the nucleon as a topological Skyrmion soliton in a nonlinear chiral field theory. By solving the static Skyrme equation numerically and quantizing rotational zero modes, it derives the nucleon and delta masses from the static energy and rotational energy contributions, yielding predictions consistent with experimental values when fitted to physical parameters.

ABSTRACT

The SU(2) Skyrme model is reviewed. The model, which considers hadron as soliton (Skyrmion), is used for investigating the nucleon mass and delta mass. Keywords: Skyrme model, soliton, hadron, nucleon mass, delta mass.

Motivation & Objective

• To investigate the nucleon and delta baryon masses within the framework of the SU(2) Skyrme model, treating hadrons as topological solitons (Skyrmions).
• To derive the static energy and moment of inertia of the Skyrmion using the Euler-Lagrange equations from the SU(2) chiral Lagrangian with Skyrme term and pion mass term.
• To quantize the collective rotational degrees of freedom of the Skyrmion and compute rotational energy contributions to the total baryon mass.
• To compare the resulting nucleon and delta masses with experimental values, validating the Skyrmion model as a viable effective field theory for low-energy hadron physics.

Proposed method

• Formulates the SU(2) Skyrme model Lagrangian with chiral symmetry, including the Skyrme term for soliton stabilization and a pion mass term.
• Derives the Euler-Lagrange equation (Skyrme equation) from the action principle, leading to a nonlinear partial differential equation for the profile function $ g(r) $.
• Solves the static Skyrme equation numerically using the Skyrme ansatz $ U = \exp(i\sigma_a \hat{r}_a g(r)) $, yielding the profile function $ g(r) $.
• Computes the static energy $ E_{\text{static}} $ and converts it to static mass $ M = E_{\text{static}} / c^2 $, which serves as the base mass of the Skyrmion.
• Quantizes the rotational zero modes by identifying SU(2) internal rotations with spatial rotations, introducing collective coordinates $ A(t) $, and computing the moment of inertia $ I $ from the profile function.
• Calculates rotational energy using $ E_{\text{rotation}} = \frac{j(j+1)}{2I} $, and combines it with the static mass to obtain the total baryon mass $ m = M + \frac{j(j+1)}{2I} $.

Experimental results

Research questions

• RQ1 Can the SU(2) Skyrme model reproduce the observed nucleon and delta baryon masses using soliton solutions of the chiral field equations?
• RQ2 What is the role of the Skyrme term in stabilizing the Skyrmion soliton against scale deformations?
• RQ3 How does the quantization of rotational zero modes contribute to the spin and mass spectrum of the nucleon and delta baryons?
• RQ4 What is the relationship between the profile function $ g(r) $, the moment of inertia $ I $, and the resulting baryon masses?

Key findings

• The static energy of the Skyrmion is found to be stable against scale transformations, satisfying both extremum and minimum stability conditions derived from scale invariance analysis.
• Numerical solution of the Skyrme equation with the Skyrme ansatz yields a profile function $ g(r) $ that is essential for computing the static mass and moment of inertia.
• The moment of inertia $ I $ is computed as a functional of $ g(r) $, with contributions from both the Skyrme term and the chiral kinetic term.
• The nucleon mass is predicted as $ m_N = M + \frac{1}{2} \cdot \frac{3}{4} \cdot \frac{1}{2I} = M + \frac{3}{16I} $, and the delta mass as $ m_\Delta = M + \frac{1}{2} \cdot \frac{15}{4} \cdot \frac{1}{2I} = M + \frac{15}{16I} $, with $ j = \frac{1}{2} $ and $ j = \frac{3}{2} $, respectively.
• The model predicts that the Skyrmion behaves as a fermion, consistent with the Wess-Zumino quantization condition requiring half-integer spin quantum numbers.
• The total baryon mass is the sum of the static mass $ M $ and the rotational energy $ \frac{j(j+1)}{2I} $, with the rotational contribution being crucial for reproducing the correct mass splitting between nucleon and delta.